Waring's Problem and the Alexander-Hirschowitz Theorem

The Big Waring Problem for polynomials asks for the smallest number *s* such that almost all homogeneous polynomial of degree *d* can be decomposed as the sum of at most *s* *d*-th powers of linear forms. This thesis sets out to describe the solution to the Big Waring Problem. We follow the proof by Brambilla and Ottaviani. We first rephrase the problem in terms of Veronese varieties, and subsequently prove the Alexander-Hirschowitz Theorem, a result in algebraic geometry that is equivalent to the Big Waring Problem. The theorem studies the behavior of Hilbert functions of collections of double points in ***P***_*n*. We then describe the Waring rank of sums of pairwise coprime monomials of fixed degree.